314:
31:
614:
108:
573:
579:
1123:
526:. The challenge is to find the refractive index as a function of radius, given that a ray describes a circular path, and further to prove the focusing properties of the lens. The solution is given in the 1854 edition of the same journal. The problems and solutions were originally published anonymously, but the solution of this problem (and one other) were included in Niven's
858:
660:
A Luneburg lens antenna offers a number of advantages over a parabolic dish. Because the lens is spherically symmetric, the antenna can be steered by moving the feed around the lens, without having to bodily rotate the whole antenna. Again, because the lens is spherically symmetric, a single lens can
597:
can be made from a
Luneburg lens by metallizing parts of its surface. Radiation from a distant radar transmitter is focussed onto the underside of the metallization on the opposite side of the lens; here it is reflected, and focussed back onto the radar station. A difficulty with this scheme is that
656:
must coincide with the point of focus, but since the phase centre is invariably somewhat inside the mouth of the horn, it cannot be brought right up against the surface of the lens. Consequently it is necessary to use a variety of
Luneburg lens that focusses somewhat beyond its surface, rather than
1717:
is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for
Maxwell's fish-eye, this first order equation can be further integrated to give a formula for
704:
For any spherically symmetric lens, each ray lies entirely in a plane passing through the centre of the lens. The initial direction of the ray defines a line which together with the centre-point of the lens identifies a plane bisecting the lens. Being a plane of symmetry of the lens, the
89:
lies at infinity, and the other on the opposite surface of the lens. J. Brown and A. S. Gutman subsequently proposed solutions which generate one internal focal point and one external focal point. These solutions are not unique; the set of solutions are defined by a set of
538:
In practice, Luneburg lenses are normally layered structures of discrete concentric shells, each of a different refractive index. These shells form a stepped refractive index profile that differs slightly from
Luneburg's solution. This kind of lens is usually employed for
1510:
425:
1118:{\displaystyle T=\int _{(r_{1},\theta _{1})}^{(r_{2},\theta _{2})}{\frac {n(r)}{c}}{\sqrt {(r\,d\theta )^{2}+dr^{2}}}={\frac {1}{c}}\int _{\theta _{1}}^{\theta _{2}}n(r){\sqrt {r^{2}+\left({\frac {dr}{d\theta }}\right)^{2}}}\,d\theta ,}
1692:
275:
755:
asserts that the path that the ray takes between them is that which it can traverse in the least possible time. Given that the speed of light at any point in the lens is inversely proportional to the refractive index, and by
300:
is the radius of the lens. Because the refractive index at the surface is the same as that of the surrounding medium, no reflection occurs at the surface. Within the lens, the paths of the rays are arcs of
1309:
1373:
77:
of two given concentric spheres onto each other. There are an infinite number of refractive-index profiles that can produce this effect. The simplest such solution was proposed by
2154:
1799:
850:
804:
661:
be used with several feeds looking in widely different directions. In contrast, if multiple feeds are used with a parabolic reflector, all must be within a small angle of the
605:
visible during training operations, or to conceal their true radar signature. Unlike other types of radar reflectors, their shape doesn't affect the handling of the aircraft.
147:
746:
194:
692:. The arrangement halves the weight of the lens, and the ground plane provides a convenient means of support. However, the feed does partially obscure the lens when the
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metallized regions block the entry or exit of radiation on that part of the lens, but the non-metallized regions result in a blind-spot on the opposite side.
202:
709:
of the refractive index has no component perpendicular to this plane to cause the ray to deviate either to one side of it or the other. In the plane, the
512:. The lens images each point on the spherical surface to the opposite point on the surface. Within the lens, the paths of the rays are arcs of circles.
522:
1540:
119:
Each point on the surface of an ideal
Luneburg lens is the focal point for parallel radiation incident on the opposite side. Ideally, the
2116:
2182:
586:
321:
Maxwell's fish-eye lens is also an example of the generalized
Luneburg lens. The fish-eye, which was first fully described by
2146:
1227:
2239:
2234:
693:
688:. This uses just one hemisphere of a Luneburg lens, with the cut surface of the sphere resting on a reflecting metal
59:
638:, but uses the lens rather than a parabolic reflector as the main focusing element. As with the dish antenna, a
673:
systems, dish antennas suffer from the feed and its supporting structure partially obscuring the main element (
1339:
1536:
757:
601:
Removable
Luneburg lens type radar reflectors are sometimes attached to military aircraft in order to make
2147:"This strange mod to the F-35 kills its stealth near Russian defenses – and there's good reason for that"
809:
763:
2057:
1876:
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149:
of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the
752:
634:
A Luneburg lens can be used as the basis of a high-gain radio antenna. This antenna is comparable to a
2029:
1947:
1912:
1213:
624:
520:
The properties of this lens are described in one of a number of set problems or puzzles in the 1853
317:
Cross-section of
Maxwell's fish-eye lens, with blue shading representing increasing refractive index
670:
644:
to the receiver or from the transmitter is placed at the focus, the feed typically consisting of a
325:
in 1854 (and therefore pre-dates
Luneburg's solution), has a refractive index varying according to
322:
125:
120:
51:
34:
Cross-section of the standard
Luneburg lens, with blue shading proportional to the refractive index
2008:"Maxwell's fisheye lens as efficient power coupler between dissimilar photonic crystal waveguides"
2045:
2019:
1963:
1821:
719:
175:
477:
is the radius of the lens's spherical surface. The index of refraction at the lens's surface is
420:{\displaystyle n(r)={\sqrt {\epsilon _{r}}}={\frac {n_{0}}{1+\left({\frac {r}{R}}\right)^{2}}},}
2178:
2124:
2077:
1217:
714:
710:
544:
2172:
1761:
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along the path of the ray. This type of minimization problem has been extensively studied in
1195:
480:
2100:
2037:
1984:
1955:
1920:
602:
150:
677:); in common with other refracting systems, the Luneburg lens antenna avoids this problem.
433:
2229:
1889:
1845:
112:
78:
1687:{\displaystyle d\theta ={\frac {h}{r{\sqrt {{\big (}n(r){\big )}^{2}r^{2}-h^{2}}}}}\,dr.}
2033:
1951:
1916:
1505:{\displaystyle n(r){\sqrt {r'^{2}+r^{2}}}-n(r){\frac {r'^{2}}{\sqrt {r'^{2}+r^{2}}}}=h,}
1314:
1781:
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1518:
1221:
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1131:
617:
594:
460:
313:
283:
155:
86:
82:
58:
decreases radially from the center to the outer surface. They can be made for use with
270:{\displaystyle n={\sqrt {\epsilon _{r}}}={\sqrt {2-\left({\frac {r}{R}}\right)^{2}}},}
2223:
1967:
666:
63:
2049:
551:
calibration standards. Cylindrical analogues of the Luneburg lens are also used for
751:
Given any two points on a ray (such as the point of entry and exit from the lens),
689:
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578:
2168:
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67:
17:
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2007:
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in 1944. Luneburg's solution for the refractive index creates two conjugate
1811:
653:
540:
107:
30:
706:
620:
91:
85:
outside the lens. The solution takes a simple and explicit form if one
1938:
Morgan, S. P. (1958). "General solution of the Luneburg lens problem".
302:
1959:
1924:
2202:
Animation of propagation through a Luneburg Lens (Dielectric Antenna)
2024:
73:
For certain index profiles, the lens will form perfect geometrical
1816:
612:
548:
312:
106:
74:
29:
2214:
Animation of a Half Maxwell's Fish-Eye Lens (Dielectric Antenna)
1852:. Providence, Rhode Island: Brown University. pp. 189–213.
1758:. In general it provides the relative rates of change of
457:
is the index of refraction at the center of the lens and
1344:
1216:, and a ready-made solution exists in the form of the
657:
the classic lens with the focus lying on the surface.
530:, which was published 11 years after Maxwell's death.
1784:
1764:
1744:
1724:
1703:
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1134:
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205:
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1985:"Solutions of problems (prob. 3, vol. VIII. p. 188)"
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1709:
1686:
1575:
1555:
1527:
1504:
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1328:
1304:{\displaystyle L(r,r')=n(r){\sqrt {r'^{2}+r^{2}}}}
1303:
1204:
1184:
1160:
1140:
1117:
844:
798:
740:
585:Luneburg reflectors (the marked protrusion) on an
504:
469:
449:
419:
292:
269:
188:
164:
141:
1824:also have a radially decreasing refractive index.
1148:is the speed of light in vacuum. Minimizing this
1903:Gutman, A. S. (1954). "Modified Luneberg Lens".
1802:to follow the path of the ray through the lens.
680:A variation on the Luneburg lens antenna is the
111:A Luneburg lens converts a point source into a
1979:
1977:
2082:The Cambridge and Dublin Mathematical Journal
1989:The Cambridge and Dublin Mathematical Journal
1635:
1615:
54:. A typical Luneburg lens's refractive index
8:
2102:The Scientific Papers of James Clerk Maxwell
2062:: CS1 maint: multiple names: authors list (
1539:. This first-order differential equation is
1224:of this second-order equation. Substituting
528:The Scientific Papers of James Clerk Maxwell
2105:. New York: Dover Publications. p. 76.
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760:, the time of transit between two points
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713:of the system makes it convenient to use
696:on the reflector is less than about 45°.
523:Cambridge and Dublin Mathematical Journal
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27:Spherically symmetric gradient-index lens
1817:BLITS (Ball Lens In The Space) satellite
1543:, that is it can be re-arranged so that
2006:Badri, S Hadi and Gilarlue, MM (2019).
1834:
1467:
1451:
1404:
1368:{\displaystyle {\tfrac {dr}{d\theta }}}
1276:
2208:Animation of a Maxwell's Fish-Eye Lens
2157:from the original on 25 December 2023.
2055:
1885:
1874:
115:when the source is placed at its edge.
669:(a form of de-focussing). Apart from
94:which must be evaluated numerically.
7:
543:, especially to construct efficient
845:{\displaystyle (r_{2},\theta _{2})}
799:{\displaystyle (r_{1},\theta _{1})}
682:hemispherical Luneburg lens antenna
748:to describe the ray's trajectory.
25:
2117:"Looking through a Luneburg Lens"
1220:, which immediately supplies the
2174:Antenna Handbook: Antenna theory
577:
571:
1629:
1623:
1563:only appears on one side, and
1441:
1435:
1394:
1388:
1266:
1260:
1251:
1234:
1172:determining the dependence of
1050:
1044:
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899:
873:
839:
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338:
1:
1850:Mathematical Theory of Optics
700:Path of a ray within the lens
630:, 1961, using a Luneburg lens
308:
142:{\displaystyle \epsilon _{r}}
50:) is a spherically symmetric
2123:. 2019-08-08. Archived from
1375:), into this identity gives
2145:Lockie, Alex (5 May 2017).
2042:10.1016/j.ijleo.2019.03.163
741:{\displaystyle (r,\theta )}
516:Publication and attribution
189:{\displaystyle {\sqrt {2}}}
2256:
686:Luneburg reflector antenna
60:electromagnetic radiation
2177:. Springer. p. 40.
1995:. Macmillan: 9–11. 1854.
102:
2088:. Macmillan: 188. 1853.
1771:{\displaystyle \theta }
1731:{\displaystyle \theta }
1576:{\displaystyle \theta }
1537:constant of integration
1205:{\displaystyle \theta }
505:{\displaystyle n_{0}/2}
309:Maxwell's fish-eye lens
1800:integrated numerically
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1732:
1711:
1688:
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1506:
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1206:
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1168:yields a second-order
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116:
35:
2171:; Lee, S. W. (1993).
2018:. Elsevier: 566–570.
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1170:differential equation
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541:microwave frequencies
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450:{\displaystyle n_{0}}
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110:
33:
1822:Gravitational lenses
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1214:Lagrangian mechanics
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196:to 1), according to
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126:
2099:Niven, ed. (1890).
2034:2019Optik.185..566B
1952:1958JAP....29.1358M
1917:1954JAP....25..855G
1583:only on the other:
1040:
934:
665:to avoid suffering
121:dielectric constant
103:Luneburg's solution
52:gradient-index lens
2240:Military deception
2235:Stealth technology
1863:Brown, J. (1953).
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1329:{\displaystyle r'}
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753:Fermat's principle
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545:microwave antennas
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92:definite integrals
36:
1960:10.1063/1.1723441
1925:10.1063/1.1721757
1884:Missing or empty
1865:Wireless Engineer
1791:{\displaystyle r}
1751:{\displaystyle r}
1738:as a function of
1710:{\displaystyle h}
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1669:
1556:{\displaystyle r}
1528:{\displaystyle h}
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1218:Beltrami identity
1185:{\displaystyle r}
1161:{\displaystyle T}
1141:{\displaystyle c}
1103:
1091:
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997:
954:
715:polar coordinates
711:circular symmetry
675:aperture blockage
609:Microwave antenna
470:{\displaystyle R}
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399:
362:
293:{\displaystyle R}
262:
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224:
184:
165:{\displaystyle n}
42:(original German
16:(Redirected from
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2151:Business Insider
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1846:Luneburg, R. K.
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1798:, which may be
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79:Rudolf Luneburg
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18:Dielectric lens
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2196:External links
2194:
2191:
2190:
2183:
2160:
2137:
2108:
2091:
2078:"Problems (3)"
2069:
1998:
1973:
1930:
1911:(7): 855–859.
1895:
1855:
1833:
1832:
1830:
1827:
1826:
1825:
1819:
1814:
1807:
1804:
1787:
1767:
1747:
1727:
1706:
1697:The parameter
1695:
1694:
1683:
1680:
1677:
1666:
1662:
1658:
1653:
1649:
1643:
1637:
1631:
1628:
1625:
1622:
1617:
1610:
1606:
1601:
1598:
1595:
1572:
1552:
1524:
1513:
1512:
1501:
1498:
1495:
1487:
1483:
1479:
1473:
1469:
1465:
1457:
1453:
1449:
1443:
1440:
1437:
1434:
1431:
1424:
1420:
1416:
1410:
1406:
1402:
1396:
1393:
1390:
1387:
1360:
1357:
1352:
1349:
1324:
1321:
1296:
1292:
1288:
1282:
1278:
1274:
1268:
1265:
1262:
1259:
1256:
1253:
1249:
1246:
1242:
1239:
1236:
1233:
1222:first integral
1201:
1181:
1157:
1137:
1126:
1125:
1114:
1111:
1108:
1100:
1095:
1089:
1086:
1081:
1078:
1072:
1067:
1062:
1058:
1052:
1049:
1046:
1043:
1036:
1032:
1024:
1020:
1015:
1009:
1006:
1001:
994:
990:
986:
983:
978:
974:
970:
967:
963:
960:
953:
949:
946:
943:
940:
932:
927:
923:
919:
914:
910:
906:
901:
896:
892:
888:
883:
879:
875:
871:
867:
864:
841:
836:
832:
828:
823:
819:
815:
795:
790:
786:
782:
777:
773:
769:
737:
734:
731:
728:
725:
701:
698:
610:
607:
584:
570:
569:
568:
567:
566:
564:
561:
535:
532:
517:
514:
501:
497:
491:
487:
466:
444:
440:
428:
427:
416:
408:
403:
398:
395:
390:
385:
382:
376:
372:
366:
359:
355:
349:
346:
343:
340:
337:
310:
307:
289:
278:
277:
266:
259:
254:
249:
246:
241:
236:
233:
228:
221:
217:
211:
208:
183:
161:
136:
132:
104:
101:
99:
96:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2252:
2241:
2238:
2236:
2233:
2231:
2228:
2227:
2225:
2215:
2212:
2209:
2206:
2203:
2200:
2199:
2195:
2186:
2184:9780442015930
2180:
2176:
2175:
2170:
2164:
2161:
2156:
2152:
2148:
2141:
2138:
2127:on 2021-09-27
2126:
2122:
2118:
2112:
2109:
2104:
2103:
2095:
2092:
2087:
2083:
2079:
2073:
2070:
2065:
2059:
2051:
2047:
2043:
2039:
2035:
2031:
2026:
2021:
2017:
2013:
2009:
2002:
1999:
1994:
1990:
1986:
1980:
1978:
1974:
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1941:
1940:J. Appl. Phys
1934:
1931:
1926:
1922:
1918:
1914:
1910:
1906:
1905:J. Appl. Phys
1899:
1896:
1891:
1878:
1870:
1866:
1859:
1856:
1851:
1847:
1841:
1839:
1835:
1828:
1823:
1820:
1818:
1815:
1813:
1810:
1809:
1805:
1803:
1801:
1785:
1765:
1745:
1725:
1704:
1681:
1678:
1675:
1664:
1660:
1656:
1651:
1647:
1641:
1626:
1620:
1608:
1604:
1599:
1596:
1593:
1586:
1585:
1584:
1570:
1550:
1542:
1538:
1522:
1499:
1496:
1493:
1485:
1481:
1477:
1471:
1463:
1455:
1447:
1438:
1432:
1429:
1422:
1418:
1414:
1408:
1400:
1391:
1385:
1378:
1377:
1376:
1358:
1355:
1350:
1347:
1322:
1319:
1294:
1290:
1286:
1280:
1272:
1263:
1257:
1254:
1247:
1244:
1240:
1237:
1231:
1223:
1219:
1215:
1199:
1179:
1171:
1155:
1135:
1112:
1109:
1106:
1098:
1093:
1087:
1084:
1079:
1076:
1070:
1065:
1060:
1056:
1047:
1041:
1034:
1030:
1022:
1018:
1013:
1007:
1004:
999:
992:
988:
984:
981:
976:
968:
965:
961:
951:
944:
938:
925:
921:
917:
912:
908:
894:
890:
886:
881:
877:
869:
865:
862:
855:
854:
853:
834:
830:
826:
821:
817:
788:
784:
780:
775:
771:
759:
754:
749:
732:
729:
726:
716:
712:
708:
699:
697:
695:
691:
687:
683:
678:
676:
672:
668:
664:
658:
655:
651:
647:
643:
642:
637:
629:
628:
622:
619:
615:
608:
606:
604:
599:
596:
588:
580:
562:
560:
558:
554:
550:
546:
542:
533:
531:
529:
525:
524:
515:
513:
499:
495:
489:
485:
464:
442:
438:
414:
406:
401:
396:
393:
388:
383:
380:
374:
370:
364:
357:
353:
347:
341:
335:
328:
327:
326:
324:
315:
306:
304:
287:
264:
257:
252:
247:
244:
239:
234:
231:
226:
219:
215:
209:
206:
199:
198:
197:
181:
159:
152:
134:
130:
122:
114:
109:
97:
95:
93:
88:
84:
80:
76:
71:
69:
65:
64:visible light
61:
57:
53:
49:
47:
41:
40:Luneburg lens
32:
19:
2216:from YouTube
2210:from YouTube
2204:from YouTube
2173:
2163:
2150:
2140:
2129:. Retrieved
2125:the original
2120:
2111:
2101:
2094:
2085:
2081:
2072:
2058:cite journal
2015:
2011:
2001:
1992:
1988:
1943:
1939:
1933:
1908:
1904:
1898:
1886:|title=
1877:cite journal
1868:
1864:
1858:
1849:
1696:
1514:
1127:
750:
703:
690:ground plane
685:
681:
679:
674:
663:optical axis
659:
650:phase centre
646:horn antenna
640:
636:dish antenna
633:
626:
600:
592:
557:laser diodes
537:
534:Applications
527:
521:
519:
429:
320:
279:
118:
72:
55:
45:
43:
39:
37:
1336:represents
555:light from
553:collimating
172:falls from
87:focal point
68:radio waves
48:neburg lens
2224:Categories
2131:2021-04-05
2025:1904.01242
1829:References
758:Pythagoras
627:Victorious
2169:Lo, Y. T.
1968:119949981
1812:Ball lens
1766:θ
1726:θ
1657:−
1597:θ
1571:θ
1541:separable
1430:−
1359:θ
1200:θ
1110:θ
1088:θ
1031:θ
1019:θ
1014:∫
969:θ
922:θ
891:θ
870:∫
831:θ
785:θ
733:θ
654:feed horn
625:HMS
354:ϵ
235:−
216:ϵ
131:ϵ
2155:Archived
2050:91184610
1848:(1944).
1806:See also
1468:′
1452:′
1405:′
1323:′
1277:′
1248:′
707:gradient
621:3D radar
618:Type 984
303:ellipses
2030:Bibcode
1948:Bibcode
1913:Bibcode
1311:(where
652:of the
323:Maxwell
98:Designs
2230:Lenses
2181:
2048:
1966:
1871:: 250.
1515:where
1128:where
671:offset
648:. The
430:where
280:where
75:images
2046:S2CID
2020:arXiv
2012:Optik
1964:S2CID
1535:is a
549:radar
62:from
2179:ISBN
2064:link
1890:help
1778:and
806:and
667:coma
641:feed
587:F-35
547:and
83:foci
2038:doi
2016:185
1956:doi
1921:doi
1192:on
852:is
684:or
623:on
66:to
2226::
2153:.
2149:.
2119:.
2084:.
2080:.
2060:}}
2056:{{
2044:.
2036:.
2028:.
2014:.
2010:.
1991:.
1987:.
1976:^
1962:.
1954:.
1944:29
1942:.
1919:.
1909:25
1907:.
1881::
1879:}}
1875:{{
1869:30
1867:.
1837:^
593:A
559:.
305:.
70:.
38:A
2187:.
2134:.
2086:8
2066:)
2052:.
2040::
2032::
2022::
1993:9
1970:.
1958::
1950::
1927:.
1923::
1915::
1892:)
1888:(
1786:r
1746:r
1705:h
1682:.
1679:r
1676:d
1665:2
1661:h
1652:2
1648:r
1642:2
1636:)
1630:)
1627:r
1624:(
1621:n
1616:(
1609:r
1605:h
1600:=
1594:d
1551:r
1523:h
1500:,
1497:h
1494:=
1486:2
1482:r
1478:+
1472:2
1464:r
1456:2
1448:r
1442:)
1439:r
1436:(
1433:n
1423:2
1419:r
1415:+
1409:2
1401:r
1395:)
1392:r
1389:(
1386:n
1356:d
1351:r
1348:d
1320:r
1295:2
1291:r
1287:+
1281:2
1273:r
1267:)
1264:r
1261:(
1258:n
1255:=
1252:)
1245:r
1241:,
1238:r
1235:(
1232:L
1180:r
1156:T
1136:c
1113:,
1107:d
1099:2
1094:)
1085:d
1080:r
1077:d
1071:(
1066:+
1061:2
1057:r
1051:)
1048:r
1045:(
1042:n
1035:2
1023:1
1008:c
1005:1
1000:=
993:2
989:r
985:d
982:+
977:2
973:)
966:d
962:r
959:(
952:c
948:)
945:r
942:(
939:n
931:)
926:2
918:,
913:2
909:r
905:(
900:)
895:1
887:,
882:1
878:r
874:(
866:=
863:T
840:)
835:2
827:,
822:2
818:r
814:(
794:)
789:1
781:,
776:1
772:r
768:(
736:)
730:,
727:r
724:(
500:2
496:/
490:0
486:n
465:R
443:0
439:n
415:,
407:2
402:)
397:R
394:r
389:(
384:+
381:1
375:0
371:n
365:=
358:r
348:=
345:)
342:r
339:(
336:n
288:R
265:,
258:2
253:)
248:R
245:r
240:(
232:2
227:=
220:r
210:=
207:n
182:2
160:n
135:r
56:n
46:ü
44:L
20:)
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